[Paper Review] The quot functor of a quasi-coherent sheaf

Gennaro Di Brino|arXiv (Cornell University)|Dec 19, 2012

Algebraic Geometry and Number Theory14 references3 citations

TL;DR

This paper constructs an infinite-dimensional scheme representing the Quot functor for quasi-coherent sheaves on a projective scheme, extending Grothendieck's classical Quot scheme beyond the coherent case. By combining Deligne's ind-object realization of quasi-coherent sheaves with a filtering inductive limit construction of the Grassmannian, the authors prove representability of the Quot functor and establish a large-scale Grassmannian embedding for uniformly regular sheaves.

ABSTRACT

We build an infinite dimensional scheme parametrizing isomorphism classes of coherent quotients of a quasi-coherent sheaf on a projective scheme. The main tool to achieve the construction is a version of Grothendieck's Grassmannian embedding combined with a result of Deligne, realizing quasi-coherent sheaves as ind-objects in the category of quasi-coherent sheaves of finite presentation. We end our treatment with the discussion of a special case in which we can retain an analog of the Grassmannian embedding.

Motivation & Objective

• To extend the classical Quot scheme construction from coherent to quasi-coherent sheaves on a projective scheme.
• To resolve the open problem of representability of the Quot functor when the sheaf is only quasi-coherent, not necessarily coherent.
• To generalize the classical Grassmannian embedding to infinite-dimensional settings using filtering inductive limits.
• To identify conditions under which the infinite-dimensional Quot scheme admits a quasi-closed embedding into a schematic Grassmannian.

Proposed method

• Utilizes Deligne's theorem that quasi-coherent sheaves on quasi-compact, quasi-separated schemes are equivalent to ind-objects of finitely presented quasi-coherent sheaves.
• Constructs the schematic Grassmannian as a filtering inductive limit of quasi-compact open subschemes via projective limits of affine morphisms between subschemes of finite-type Grassmannians.
• Represents the Quot functor as a filtering inductive limit of schemes obtained as projective limits of open subschemes of classical Quot schemes and affine morphisms.
• Applies the ind-pro principle to build the infinite-dimensional Quot scheme as a limit of finite-dimensional Quot schemes.
• Introduces the notion of uniformly m-regular quasi-coherent sheaves, where all coherent approximations have Castelnuovo-Mumford regularity bounded by m.
• Establishes a quasi-closed embedding of the Quot scheme into a Grassmannian of a direct limit of global sections, using compatibility of the embedding with the filtering structure.

Experimental results

Research questions

• RQ1 Can the Quot functor for a quasi-coherent sheaf on a projective scheme be represented by a scheme, even when the sheaf is not coherent?
• RQ2 Is there a generalization of the classical Grassmannian embedding for the Quot scheme in the infinite-dimensional setting?
• RQ3 Under what conditions on a quasi-coherent sheaf does the Quot scheme admit a closed embedding into a schematic Grassmannian?
• RQ4 How can the ind-object structure of quasi-coherent sheaves be used to construct the Quot scheme as a filtered inductive limit?
• RQ5 What role does Castelnuovo-Mumford regularity play in controlling the geometry of the Quot scheme for quasi-coherent sheaves?

Key findings

• The Quot functor for a quasi-coherent sheaf on a projective scheme over a field is representable by a scheme, generalizing Grothendieck's result from the coherent case.
• The Quot scheme is constructed as a filtering inductive limit of schemes, each arising as a projective limit of open subschemes of classical Quot schemes and affine morphisms.
• The schematic Grassmannian of locally free quotients of a quasi-coherent sheaf is shown to be a filtering inductive limit of quasi-compact open subschemes.
• For uniformly m-regular quasi-coherent sheaves, there exists a quasi-closed embedding of the Quot scheme into a Grassmannian of a direct limit of global sections of twists of the sheaf.
• The embedding is realized as a limit of closed embeddings between components of the filtered systems, preserving the ind-pro structure.
• The construction recovers the classical projective bundle as a special case when the Hilbert polynomial is constant 1, identifying the Quot scheme with the projectivization of the sheaf.